Let A denote a finite-dimensional algebra over a field k and suppose X is a projective variety over k, which is not a curve of genus 0 or 1. It is shown that there exists a vector bundle F over X whose endomorphism algebra is isomorphic to A. An analogous theorem is proven for vector bundles with symmetric or anti-symmetric regular bilinear forms. In both cases a vector bundle with the desired endomorphism algebra is explicitly constructed as an extension of direct sums of suitable vector bundles. The results are also expressed in terms of category theory.
Reviewer: H.Lange (Erlangen)